We study the appication of the conjugated gradient method preconditioned by a limited memory quasi- Newton matrix in the resolution of the FAIPA’s internal linear systems.
FAIPA, the Feasible Arc Interior Point Algorithm,[[
]], is an interior-point algorithm that solves nonlinear optimization problems. It makes iterations in the primal and dual variables of the optimization problem to solve the Karush-Kuhn-Tucker optimality conditions. Given an initial interior point, it defines a sequence of interior points with the objetive function monotonically reduced.
FAIPA requires the solution of three linear systems whit the same coefficient matrix at each iteration. These systems, when solved in terms of the Lagrange’s multipliers, are in general full, symmetric and positive definite. The conjugated gradient is largely employed for the iterative solution of symmetric and positive definite linear systems.
When the system is badly conditioned, a preconditioner matrix con be employed. The preconditioner’s choice is fundamental for the technique’s efficiency.
We present a preconditioner for the FAIPA’s linear systems based on the limited memoy BFGS technique and consider problems with full and sparse matrices. Numerical examples show the performance of our methodology.